Determinant nonvanishing implies local invertibility lemma

Let $A:\mathbb{R}^n\to\mathbb{R}^n$ be a linear map. Saying $\det A\neq 0$ is equivalent to saying $A$ is invertible.

Stability of invertibility (Neumann series lemma): If $A$ is invertible and $B$ is another linear map such that

\|A^{-1}(B-A)\|<1,

then $B$ is invertible and

B^{-1}=\sum_{k=0}^\infty \bigl(-A^{-1}(B-A)\bigr)^k\,A^{-1}.

Moreover,

\|B^{-1}\|\le \frac{\|A^{-1}\|}{1-\|A^{-1}(B-A)\|}.

In particular, if $\|B-A\|\le \frac{1}{2\|A^{-1}\|}$ then $B$ is invertible and $\|B^{-1}\|\le 2\|A^{-1}\|$ .

Remarks

This lemma is a key linear-algebraic ingredient in the inverse function theorem: once $Df(a)$ is invertible, $Df(x)$ remains invertible for all $x$ sufficiently close to $a$ (because $Df$ is continuous).